Problem: What is the period of $y=8\cos\left(5\pi x+\dfrac{3\pi}{2}\right)-9$ ? Give an exact value. units
Explanation: Period in sinusoids of the form $y=a\cos(bx+c)+d$ Graphically, the period of a sinusoidal function is the horizontal distance between the ends of a single cycle of its graph. The period of a sinusoid of the form $y={a}\cos( bx + c) + {d}$ is equal to $\dfrac{2\pi}{| b|}$. [How can we justify this given our graphical understanding of period?] Finding the period The period of $y = 8\cos\left({5\pi}x+\dfrac{3\pi}{2}\right)-9$ is: $\begin{aligned} \text{period}&=\dfrac{2\pi}{|{b}|}\\\\ &=\dfrac{2\pi}{|{5\pi}|} \\\\\\\\\\ &= \dfrac{2}{5} \\ \end{aligned}$ The answer The period of $y = 8\cos\left({5\pi}x+\dfrac{3\pi}{2}\right)-9$ is $\dfrac25$ units.